Strichartz estimates for $N$-body Schrödinger operators with small potential interactions

Younghun Hong

doi:10.3934/dcds.2017233

Article Contents

2017, Volume 37, Issue 10: 5355-5365. Doi: 10.3934/dcds.2017233

This issue Previous Article On the Bonsall cone spectral radius and the approximate point spectrum Next Article Multifractal analysis of random weak Gibbs measures

Strichartz estimates for $N$-body Schrödinger operators with small potential interactions

Younghun Hong^,

Center for Mathematical Analysis & Computation (CMAC), Yonsei University, Seoul 03722, Korea

* Corresponding author: Younghun Hong
* Corresponding author: Younghun Hong

Received: January 2016

Revised: May 2017

Published: October 2017

The author is supported by NRF grant 2015R1A5A1009350.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we prove Strichartz estimates for $N$-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the $V_S^p$-norm of Koch and Tataru [19]. As an application, we prove scattering for $N$-body Schrödinger operators.

Keywords:

Mathematics Subject Classification: Primary: 35Q40, 35Q41, 81U10; Secondary: 35B40, 35Q70.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	M. Beceanu and M. Goldberg , Schrödinger dispersive estimates for a scaling-critical class of potentials, Comm. Math. Phys., 314 (2012) , 471-481. doi: 10.1007/s00220-012-1435-x.
	J. Bergh and J. Löfström, Interpolation Spaces – An Introduction, Grundlehren der Mathematischen Wissenschaften, 1976, x+207 pp.
	N. Burq , F. Planchon , J. Stalker and S. Tahvildar-Zadeh , Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003) , 519-549. doi: 10.1016/S0022-1236(03)00238-6.
	N. Burq , F. Planchon , J. Stalker and S. Tahvildar-Zadeh , Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004) , 1665-1680. doi: 10.1512/iumj.2004.53.2541.
	T. Chen , Y. Hong and N. Pavlović , Global well-posedness of the NLS system for infinitely many fermions, Arch. Ration. Mech. Anal., 224 (2017) , 91-123. doi: 10.1007/s00205-016-1068-x.
	T. Chen and N. Pavlović , Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in d=3 based on spacetime norms, Ann. Henri Poincaré, 15 (2014) , 543-588. doi: 10.1007/s00023-013-0248-6.
	X. Chen and J. Holmer , On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (JEMS), 18 (2016) , 1161-1200. doi: 10.4171/JEMS/610.
	D. Fujiwara , Remarks on convergence of the Feynman path integrals, Duke Math. J., 47 (1980) , 559-600. doi: 10.1215/S0012-7094-80-04734-1.
	M. Goldberg , Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials, Amer. J. Math., 128 (2006) , 731-750. doi: 10.1353/ajm.2006.0025.
	M. Goldberg , Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials, Geom. Funct. Anal., 16 (2006) , 517-536. doi: 10.1007/s00039-006-0568-5.
	M. Goldberg and W. Schlag , Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004) , 157-178. doi: 10.1007/s00220-004-1140-5.
	G. Graf , Asymptotic completeness for $N$-body short-range quantum systems: A new proof, Comm. Math. Phys., 132 (1990) , 73-101. doi: 10.1007/BF02278000.
	M. Hadac , S. Herr and H. Koch , Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009) , 917-941. doi: 10.1016/j.anihpc.2008.04.002.
	S. Herr , D. Tataru and N. Tzvetkov , Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${{H}^{\text{1}}}\left( {{\mathbb{T}}^{\text{3}}} \right)$, Duke Math. J., 159 (2011) , 329-349. doi: 10.1215/00127094-1415889.
	J.-L. Journe , A. Soffer and C. Sogge , Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991) , 573-604. doi: 10.1002/cpa.3160440504.
	M. Keel and T. Tao , Endpoint Strichartz estimates, Amer. J. Math., 120 (1998) , 955-980. doi: 10.1353/ajm.1998.0039.
	K. Kirkpatrick , B. Schlein and G. Staffilani , Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133 (2011) , 91-130. doi: 10.1353/ajm.2011.0004.
	S. Klainerman and M. Machedon , On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008) , 169-185. doi: 10.1007/s00220-008-0426-4.
	H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces Int. Math. Res. Not. , 16 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.
	H. Koch, D. Tataru and M. Vişan, Dispersive equations and nonlinear waves: Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps Oberwolfach Seminars, 45 (2014). doi: 10.1007/978-3-0348-0736-4.
	I. Rodnianski and W. Schlag , Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004) , 451-513. doi: 10.1007/s00222-003-0325-4.
	W. Schlag, Dispersive estimates for Schrödinger operators: A survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., Princeton Univ. Press, 163 (2007), 255–285.
	I. M. Sigal and A. Soffer , The N-particle scattering problem: Asymptotic completeness for short-range systems, Ann. of Math., 126 (1987) , 35-108. doi: 10.2307/1971345.